3. GRAVITATIONAL LENS OPTICS

Although basic geometrical optics is all that is required to describe
gravitational lenses, the development of the subject has generated
some fresh physical insights. Several complementary formalisms have
been derived (cf
Nityananda 1990a,
b),
but we shall confine our
attention to those that are most appropriate for application to
cosmology.
Schneider et
al. (1992)
provide a more complete treatment of the subject.

Figure 5. Basic ray geometry of gravitational
lensing. A
light ray from a source S at redshift zs is
incident on a deflector or lens L at redshift
zd with impact parameter relative to some
fiducial lens ``center''. Assuming the lens is thin compared to the
total path length, its influence can be described by a deflection
angle dot () (a two-vector)
suffered by the ray on crossing the ``lens plane''. The deflection ray
reaches the observer O, who sees the image of the source
apparently at position
on the sky. The true direction of the source, i.e. its position on the
sky in the absence of the lens, is indicated by . Also shown are
the angular diameter distances Dd,
Ds, Dds, separating the source,
deflector, and observer.

The basic ray geometry is shown in Figure 5.
For the moment, consider a homogeneous
Friedmann-Robertson-Walker (FRW) universe of cosmological density
parameter 0 in
which angular diameter distancesD (zi, zj) relate proper lengths j
located at redshift zj
to angles i
subtended by these lengths when observed from
redshift zi < zj:

1.

where H0 = 100h km s-1
Mpc-1 is the Hubble constant. For
a source at redshift zs and a single deflector at redshift
zd,
we follow convention and set Dd = D(0,
zd), Ds = D (0,
zs), and Dds =
D (zd, zs). Elementary geometry
gives the lens equation (e.g.
Refsdal 1964a),
which connects the
source position and the image position (Figure 5),

2.

where the reduced deflection angle () and
the true deflection angle hat ()
are related by

Deflection angles of interest in astrophysical applications are always
small. We are therefore justified in using the weak field
approximation of General Relativity. For a thin lens,
hat = / c2,
where () is twice the
2D Newtonian potential obtained by solving the 2D Poisson
equation, 2() = 8G(),
corresponding to surface mass density (). [A weak gravitational lens is, in its essentials,
equivalent to a flat space
deflector with refractive index 1 - 2/ c2, where is
the 3D Newtonian potential with respect to infinity
(Eddington 1919).]

From the structure of Equation 2 it is clear that, for a given lens,
there is a unique source position for each image
position
. However, the converse
is not true. For a nontrivial
deflection law (), it is possible to find more
than one solution
satisfying Equation 2 for a given
, thus giving
rise to multiple imaging. (In
general, gravitational lenses are not like simple optical lenses which
have hat () varying linearly with
.)

The magnification tensor of the ith image relative to the
unlensed source is

3.

which is a symmetric 2 x 2 matrix. This is not directly measurable,
but the relative magnification between two images, described by
the transformation matrix [µ(ij)] =
[µ(i)] [µ(j)]-1
can be estimated when the images are resolved. This is in general
nonsymmetric and provides four independent observables. The flux
magnification associated with the ith image is the absolute
value of the determinant of the magnification tensor,
i.e. µ(i) = | [µ(i)] |. For
unresolved images, only the relative flux
magnifications, µij =
| [µ(ij)] |, are measurable.